of a Surface of the Second Order into itself. 329 



#2> Vv z v w 9. linear functions of x lt y x , z v w u such that 



Write 



<Bi + * % *k%% yi + tj 2 = 2v, *,+s 8 = 2?, w l + w^2a); 



then putting # 2 =2|f--<z , 1 , &c, the proposed equation will be 

 satisfied if only 



f 2 + ^+? 2 + ^ = ^H-Wi + ^i+^i J 

 which will obviously be the case if 



y x = — vf-f ?? + Xf+6&> 



2*j= fig—XT)-}- f +CQ) 



w 1 =: —ag—bij — cZ+G), 



where X, /&, v, a, b, c are arbitrary. 

 Write for shortness 



aX + bjjL-\- cv = (f> 

 l + X 2 + ^ 2 + ^ + « 2 +6 2 + ^-f^ 2 = ^ 

 then we have 



ki; = (1 + X 2 + b 2 + c 2 )^ -f (X/*— v — ab — c<j>)i/ l + (vX +/* — ca + £<£).?, 



+ (6 v — c/x — a — X(j>) W\ 

 krj = (\/a + v — ab + c$)x x + (1 + y^ 2 -f c 2 + a 2 )^ + (fiv — X— 6c 4- a$)z x 



+ (c\ — av—b—/jL(f>)w l 

 kZ=(vX—fz — ca — b(j>)a; x + (/jLv + X--bc + a<p)i/ x -|- (1 + v 2 + « 2 -f 6 2 )5' 1 



+ (ayu, — Z>X — - c — v</>) w x 

 Z; w ss (bv — c/i, + « + Xfy)x x + (cX — av + 6 -f- /a^)^ + (a/A — 6v + c + p^)#i 



+ (l+X 2 H-A6 2 + v> i; 

 and from these we obtain at ones 

 ^ 2 ==(l+X 2 + 6 2 + c 2 --yu, 2 -v 2 --« 2 -(^> 2 )^ + 2(X^--v--^~^)?/ l 



+ 2 (vX -f fju — c# + b(p)z x + 2 (dr — <?/u. — a — Xcf>)ia 1 

 % 2 = 2(X^ + v-«6 + ^)^ 1 + (l+^ 2 + c 2 + « 2 -v 2 -X 2 -6 2 -0 2 )?/ 1 



+ 2 (/Ltv — X — be — axf))z l + 2(eX— av — b — fi<p)w 1 

 kz 2 —2(vX—/jb — ca — b<j))x l + 2(/xv + X— be + «<£)?/! 



4-(l + v 2 + « 2 +6 2 -X 2 -^ 2 ~c 2 -(/) 2 )^ 1 +2(^-Z'X-c~^)w 1 

 kw 2 =2(bv—c/ju + a + X^)^ + 2(cX — «v -f 6 4- /^>)?/i 



+ 2(^~6v + c + v</>> 1 + (l4-X 2 + ^ 2 + v 2 -^-6 2 -c 2 ~^> 2 )w; 1 , 



values which satisfy identically 



** 2 + vi + * 2 2 + < = *? + 2/i 2 + *i 2 + <• 

 Pfo7. Mag. S. 4. Vol. 6. No. 40. iVbv. 1853. Z 



